Our MAV conference workshop on hands-on problem solving tasks is full, an indication perhaps that teachers are exploring whether this approach can make a significant contribution to addressing the problem solving and reasoning proficiencies of the Australian curriculum.

Also 4 schools have invested in the eTask Package in the past month - 2 secondary schools and 2 primary schools; 2 of the schools being in Queensland, 1 in Victoria and 1 in New Zealand.

More than one of the contact teachers in these 4 schools expressed excitement at working with the staff to create as much of their Task Library as possible ready for the next school year. Welcome and best wishes to all of you as you begin this new professional learning adventure.

That's a total of 65 schools, institutions and, occasionally, individual teachers who have invested in this resource since it was released in June 2017. Given 40 weeks in a school year, about one each school week on average since then. That's a significant number moving towards making learning to work like a mathematician the core pathway for achieving curriculum expectations.

This task appears to defy the concept of conservation of area. The same four pieces are placed on a grid in two different ways and appear to make the same triangle and cover the same area. Yet in one arrangement there is a hole in the middle of the triangle exactly equal to one square unit - one extra square of area! It's not possible, but how do we convince ourselves that what we think we see isn't really what is there?

As a puzzle it's a great activity at any level where the students understand conservation of area. In middle high school it's a fabulous task because the disproving of what we think we see depends on gradient, co-ordinate geometry, area of a triangle and the concept of proof. The task also includes extensions based on Fibonacci and the Golden Ratio.

In the eTask Package this task is in the 'more work' set because it requires extra time to make the 4 pieces from the template supplied. They can be made by printing, laminating and cutting, or carefully crafting from wood or plastic. The template also includes a matching grid.

Task 240, Less Than Fractions

So easy to set up. Lots of early success and then, wham!, how can we find all the solutions.

We have one set of tiles numbered 1 to 9. We place one on top of another to represent a fraction. How many ways can we do this to make a fraction that is less than one.

Pretty much every pair of students from Year 2 on will find some solutions. But how do we find them all?

And how do we find solutions when the problem extends to finding the two fractions which sum to less than 1, still using the tiles 1 to 9?

One way might be to let the first fraction be one half. Then the second one must be less than one half if the total is to be less than one.
This encourages students to want to know if a fraction is less than a half and that helps them return to the fraction concept - a whole divided into equal parts - to decide which ways of dividing will make make combinations of equal parts that are less than 1/2.

In the eTask Package this task is in the 'easy to make' set because it only needs 9 numbered tiles which fit the square on the card.

You can search for lessons by Year Level, Curriculum Strand, Lesson Features & Keyword at Maths300.

You can connect Tasks to their Maths300 companion lesson using Tasks & Maths300. (PDF file) Excel version provided by Pymble Ladies' College also links each Task to its Maths With Attitude content strand.